It is well-known that overparametrized neural networks trained using gradient-based methods quickly achieve small training error with appropriate hyperparameter settings. Recent papers have proved this statement theoretically for highly overparametrized networks under reasonable assumptions. These results either assume that the activation function is ReLU or they crucially depend on the minimum eigenvalue of a certain Gram matrix depending on the data, random initialization and the activation function. In the later case, existing works only prove that this minimum eigenvalue is non-zero and do not provide quantitative bounds. On the empirical side, a contemporary line of investigations has proposed a number of alternative activation functions which tend to perform better than ReLU at least in some settings but no clear understanding has emerged. This state of affairs underscores the importance of theoretically understanding the impact of activation functions on training. In the present paper, we provide theoretical results about the effect of activation function on the training of highly overparametrized 2-layer neural networks. A crucial property that governs the performance of an activation is whether or not it is smooth. For non-smooth activations such as ReLU, SELU and ELU, all eigenvalues of the associated Gram matrix are large under minimal assumptions on the data. For smooth activations such as tanh, swish and polynomials, the situation is more complex. If the subspace spanned by the data has small dimension then the minimum eigenvalue of the Gram matrix can be small leading to slow training. But if the dimension is large and the data satisfies another mild condition, then the eigenvalues are large. If we allow deep networks, then the small data dimension is not a limitation provided that the depth is sufficient. We discuss a number of extensions and applications of these results.
Duke University（杜克大学）；IIIS, Tsinghua University（清华大学交叉信息研究院）
It has been observed \citep{zhang2016understanding} that deep neural networks can memorize: they achieve 100\% accuracy on training data. Recent theoretical results explained such behavior in highly overparametrized regimes, where the number of neurons in each layer is larger than the number of training samples. In this paper, we show that neural networks can be trained to memorize training data perfectly in a mildly overparametrized regime, where the number of parameters is just a constant factor more than the number of training samples, and the number of neurons is much smaller.
Neural networks have become a prominent approach to solve inverse problems in recent years. Amongst the different existing methods, the Deep Image/Inverse Priors (DIPs) technique is an unsupervised approach that optimizes a highly overparametrized neural network to transform a random input into an object whose image under the forward model matches the observation. However, the level of overparametrization necessary for such methods remains an open problem. In this work, we aim to investigate this question for a two-layers neural network with a smooth activation function. We provide overparamet…
We propose convex relaxations for convolutional neural nets with one hidden layer where the output weights are fixed. For convex activation functions such as rectified linear units, the relaxations are convex second order cone programs which can be solved very efficiently. We prove that the relaxation recovers the global minimum under a planted model assumption, given sufficiently many training samples from a Gaussian distribution. We also identify a phase transition phenomenon in recovering the global minimum for the relaxation.
Overwhelming theoretical and empirical evidence shows that mildly overparametrized neural networks -- those with more connections than the size of the training data -- are often able to memorize the training data with $100\%$ accuracy. This was rigorously proved for networks with sigmoid activation functions and, very recently, for ReLU activations. Addressing a 1988 open question of Baum, we prove that this phenomenon holds for general multilayered perceptrons, i.e. neural networks with threshold activation functions, or with any mix of threshold and ReLU activations. Our construction is prob…
Michigan Technological University（密歇根理工大学）；Symbiosis Centre for Medical Image Analysis（共生医学图像分析中心）；Symbiosis Institute of Technology（共生技术学院）
This work introduces a novel activation unit that can be efficiently employed in deep neural nets (DNNs) and performs significantly better than the traditional Rectified Linear Units (ReLU). The function developed is a two parameter version of the specialized Richard's Curve and we call it Adaptive Richard's Curve weighted Activation (ARiA). This function is non-monotonous, analogous to the newly introduced Swish, however allows a precise control over its non-monotonous convexity by varying the hyper-parameters. We first demonstrate the mathematical significance of the two parameter ARiA follo…
Imperial College London（帝国理工学院）；Department of Computing, Imperial College London, UK（帝国理工学院计算机系）
Deep Convolutional Neural Networks (DCNNs) is currently the method of choice both for generative, as well as for discriminative learning in computer vision and machine learning. The success of DCNNs can be attributed to the careful selection of their building blocks (e.g., residual blocks, rectifiers, sophisticated normalization schemes, to mention but a few). In this paper, we propose $Π$-Nets, a new class of DCNNs. $Π$-Nets are polynomial neural networks, i.e., the output is a high-order polynomial of the input. $Π$-Nets can be implemented using special kind of skip connections and their par…
Estimation of functions of $ d $ variables is considered using ridge combinations of the form $ \textstyle\sum_{k=1}^m c_{1,k} ϕ(\textstyle\sum_{j=1}^d c_{0,j,k}x_j-b_k) $ where the activation function $ ϕ$ is a function with bounded value and derivative. These include single-hidden layer neural networks, polynomials, and sinusoidal models. From a sample of size $ n $ of possibly noisy values at random sites $ X \in B = [-1,1]^d $, the minimax mean square error is examined for functions in the closure of the $ \ell_1 $ hull of ridge functions with activation $ ϕ$. It is shown to be of order $ …
University of California, Irvine（加州大学欧文分校）；University at Albany, State University of New York（纽约州立大学奥尔巴尼分校）
Quantized or low-bit neural networks are attractive due to their inference efficiency. However, training deep neural networks with quantized activations involves minimizing a discontinuous and piecewise constant loss function. Such a loss function has zero gradients almost everywhere (a.e.), which makes the conventional gradient-based algorithms inapplicable. To this end, we study a novel class of \emph{biased} first-order oracle, termed coarse gradient, for overcoming the vanished gradient issue. A coarse gradient is generated by replacing the a.e. zero derivatives of quantized (i.e., stair-c…
This paper shows that every sublevel set of the loss function of a class of deep over-parameterized neural nets with piecewise linear activation functions is connected and unbounded. This implies that the loss has no bad local valleys and all of its global minima are connected within a unique and potentially very large global valley.
The Chinese University of Hong Kong（香港中文大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Does over-parameterization eliminate sub-optimal local minima for neural networks? An affirmative answer was given by a classical result in [59] for 1-hidden-layer wide neural networks. A few recent works have extended the setting to multi-layer neural networks, but none of them has proved every local minimum is global. Why is this result never extended to deep networks? In this paper, we show that the task is impossible because the original result for 1-hidden-layer network in [59] can not hold. More specifically, we prove that for any multi-layer network with generic input data and non-linea…
We present a new class of neurons, ARNs, which give a cross entropy on test data that is up to three times lower than the one achieved by carefully optimized LSTM neurons. The explanations for the huge improvements that often are achieved are elaborate skip connections through time, up to four internal memory states per neuron and a number of novel activation functions including small quadratic forms. The new neurons were generated using automatic programming and are formulated as pure functional programs that easily can be transformed. We present experimental results for eight datasets and fo…
University of Colorado, Boulder（科罗拉多大学博尔德分校）；Google Research（谷歌研究院）
In human perception and cognition, a fundamental operation that brains perform is interpretation: constructing coherent neural states from noisy, incomplete, and intrinsically ambiguous evidence. The problem of interpretation is well matched to an early and often overlooked architecture, the attractor network---a recurrent neural net that performs constraint satisfaction, imputation of missing features, and clean up of noisy data via energy minimization dynamics. We revisit attractor nets in light of modern deep learning methods and propose a convolutional bipartite architecture with a novel t…
Center for Machine Perception（机器感知中心）；Czech Technical University in Prague（捷克布拉格技术大学）
Layer-sequential unit-variance (LSUV) initialization - a simple method for weight initialization for deep net learning - is proposed. The method consists of the two steps. First, pre-initialize weights of each convolution or inner-product layer with orthonormal matrices. Second, proceed from the first to the final layer, normalizing the variance of the output of each layer to be equal to one. Experiment with different activation functions (maxout, ReLU-family, tanh) show that the proposed initialization leads to learning of very deep nets that (i) produces networks with test accuracy better or…
Improving the accuracy and robustness of deep neural nets (DNNs) and adapting them to small training data are primary tasks in deep learning research. In this paper, we replace the output activation function of DNNs, typically the data-agnostic softmax function, with a graph Laplacian-based high dimensional interpolating function which, in the continuum limit, converges to the solution of a Laplace-Beltrami equation on a high dimensional manifold. Furthermore, we propose end-to-end training and testing algorithms for this new architecture. The proposed DNN with graph interpolating activation i…
Deep neural networks are currently among the most commonly used classifiers. Despite easily achieving very good performance, one of the best selling points of these models is their modular design - one can conveniently adapt their architecture to specific needs, change connectivity patterns, attach specialised layers, experiment with a large amount of activation functions, normalisation schemes and many others. While one can find impressively wide spread of various configurations of almost every aspect of the deep nets, one element is, in authors' opinion, underrepresented - while solving clas…
In recent years, deep neural networks (DNNs) achieved unprecedented performance in many low-level vision tasks. However, state-of-the-art results are typically achieved by very deep networks, which can reach tens of layers with tens of millions of parameters. To make DNNs implementable on platforms with limited resources, it is necessary to weaken the tradeoff between performance and efficiency. In this paper, we propose a new activation unit, which is particularly suitable for image restoration problems. In contrast to the widespread per-pixel activation units, like ReLUs and sigmoids, our un…
Recently, fully-connected and convolutional neural networks have been trained to achieve state-of-the-art performance on a wide variety of tasks such as speech recognition, image classification, natural language processing, and bioinformatics. For classification tasks, most of these "deep learning" models employ the softmax activation function for prediction and minimize cross-entropy loss. In this paper, we demonstrate a small but consistent advantage of replacing the softmax layer with a linear support vector machine. Learning minimizes a margin-based loss instead of the cross-entropy loss. …
Qualcomm AI Research, Qualcomm Technologies, Inc.（高通AI研究，高通技术公司）；Qualcomm AI Research, Qualcomm Technologies Netherlands B.V.（高通AI研究，高通技术荷兰有限公司）；Seoul National University（首尔大学）
Unlike ReLU, newer activation functions (like Swish, H-swish, Mish) that are frequently employed in popular efficient architectures can also result in negative activation values, with skewed positive and negative ranges. Typical learnable quantization schemes [PACT, LSQ] assume unsigned quantization for activations and quantize all negative activations to zero which leads to significant loss in performance. Naively using signed quantization to accommodate these negative values requires an extra sign bit which is expensive for low-bit (2-, 3-, 4-bit) quantization. To solve this problem, we prop…
The problem we address is the following: how can a user employ a predictive model that is held by a third party, without compromising private information. For example, a hospital may wish to use a cloud service to predict the readmission risk of a patient. However, due to regulations, the patient's medical files cannot be revealed. The goal is to make an inference using the model, without jeopardizing the accuracy of the prediction or the privacy of the data. To achieve high accuracy, we use neural networks, which have been shown to outperform other learning models for many tasks. To achieve t…
University of California, Los Angeles（加州大学洛杉矶分校）；Hong Kong University of Science and Technology（香港科技大学）；Duke University（杜克大学）；Tsinghua University（清华大学）
We replace the output layer of deep neural nets, typically the softmax function, by a novel interpolating function. And we propose end-to-end training and testing algorithms for this new architecture. Compared to classical neural nets with softmax function as output activation, the surrogate with interpolating function as output activation combines advantages of both deep and manifold learning. The new framework demonstrates the following major advantages: First, it is better applicable to the case with insufficient training data. Second, it significantly improves the generalization accuracy o…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；RWTH-Aachen University（亚琛工业大学）；Simula Research Laboratory（西穆拉研究实验室）；University of San Diego（圣地亚哥大学）
In this paper we approach the problem of unique and stable identifiability of generic deep artificial neural networks with pyramidal shape and smooth activation functions from a finite number of input-output samples. More specifically we introduce the so-called entangled weights, which compose weights of successive layers intertwined with suitable diagonal and invertible matrices depending on the activation functions and their shifts. We prove that entangled weights are completely and stably approximated by an efficient and robust algorithm as soon as $\mathcal O(D^2 \times m)$ nonadaptive inp…
Deploying deep learning models, comprising of non-linear combination of millions, even billions, of parameters is challenging given the memory, power and compute constraints of the real world. This situation has led to research into model compression techniques most of which rely on suboptimal heuristics and do not consider the parameter redundancies due to linear dependence between neuron activations in overparametrized networks. In this paper, we propose a novel model compression approach based on exploitation of linear dependence, that compresses networks by elimination of entire neurons an…
University of Chicago Booth School of Business（芝加哥大学布斯商学院）
We study deep neural networks and their use in semiparametric inference. We establish novel rates of convergence for deep feedforward neural nets. Our new rates are sufficiently fast (in some cases minimax optimal) to allow us to establish valid second-step inference after first-step estimation with deep learning, a result also new to the literature. Our estimation rates and semiparametric inference results handle the current standard architecture: fully connected feedforward neural networks (multi-layer perceptrons), with the now-common rectified linear unit activation function and a depth ex…
In this paper, we propose majority voting neural networks for sparse signal recovery in binary compressed sensing. The majority voting neural network is composed of several independently trained feedforward neural networks employing the sigmoid function as an activation function. Our empirical study shows that a choice of a loss function used in training processes for the network is of prime importance. We found a loss function suitable for sparse signal recovery, which includes a cross entropy-like term and an $L_1$ regularized term. From the experimental results, we observed that the majorit…
LIX, Ecole Polytechnique, CNRS and Institut Polytechnique de Paris（巴黎综合理工学院与巴黎综合理工学院联盟计算机科学实验室）；University of Colorado at Boulder（科罗拉多大学博尔德分校）
This paper studies the problem of range analysis for feedforward neural networks, which is a basic primitive for applications such as robustness of neural networks, compliance to specifications and reachability analysis of neural-network feedback systems. Our approach focuses on ReLU (rectified linear unit) feedforward neural nets that present specific difficulties: approaches that exploit derivatives do not apply in general, the number of patterns of neuron activations can be quite large even for small networks, and convex approximations are generally too coarse. In this paper, we employ set-…
Department of Mathematics, Texas A\&M, College Station, United States
This article concerns the expressive power of depth in neural nets with ReLU activations and bounded width. We are particularly interested in the following questions: what is the minimal width $w_{\text{min}}(d)$ so that ReLU nets of width $w_{\text{min}}(d)$ (and arbitrary depth) can approximate any continuous function on the unit cube $[0,1]^d$ aribitrarily well? For ReLU nets near this minimal width, what can one say about the depth necessary to approximate a given function? Our approach to this paper is based on the observation that, due to the convexity of the ReLU activation, ReLU nets a…
Princeton University（普林斯顿大学）；Institute for Advanced Study（高等研究院）
Adversarial training is a popular method to give neural nets robustness against adversarial perturbations. In practice adversarial training leads to low robust training loss. However, a rigorous explanation for why this happens under natural conditions is still missing. Recently a convergence theory for standard (non-adversarial) supervised training was developed by various groups for {\em very overparametrized} nets. It is unclear how to extend these results to adversarial training because of the min-max objective. Recently, a first step towards this direction was made by Gao et al. using too…
We trained three Binarized Convolutional Neural Network architectures (LeNet-4, Network-In-Network, AlexNet) on a variety of datasets (MNIST, CIFAR-10, CIFAR-100, extended SVHN, ImageNet) using error-prone activations and tested them without errors to study the resilience of the training process. With the exception of the AlexNet when trained on the ImageNet dataset, we found that Bit Error Rates of a few percent during training do not degrade the test accuracy. Furthermore, by training the AlexNet on progressively smaller subsets of ImageNet classes, we observed increasing tolerance to activa…
The generalization mystery of overparametrized deep nets has motivated efforts to understand how gradient descent (GD) converges to low-loss solutions that generalize well. Real-life neural networks are initialized from small random values and trained with cross-entropy loss for classification (unlike the "lazy" or "NTK" regime of training where analysis was more successful), and a recent sequence of results (Lyu and Li, 2020; Chizat and Bach, 2020; Ji and Telgarsky, 2020) provide theoretical evidence that GD may converge to the "max-margin" solution with zero loss, which presumably generalize…
Jiangxi Normal University（江西师范大学）；City University of Hong Kong（香港城市大学）；Hong Kong University of Science and Technology（香港科技大学）
In this paper, we develop an alternating direction method of multipliers (ADMM) for deep neural networks training with sigmoid-type activation functions (called \textit{sigmoid-ADMM pair}), mainly motivated by the gradient-free nature of ADMM in avoiding the saturation of sigmoid-type activations and the advantages of deep neural networks with sigmoid-type activations (called deep sigmoid nets) over their rectified linear unit (ReLU) counterparts (called deep ReLU nets) in terms of approximation. In particular, we prove that the approximation capability of deep sigmoid nets is not worse than t…
This theoretical paper is devoted to developing a rigorous theory for demystifying the global convergence phenomenon in a challenging scenario: learning over-parameterized Rectified Linear Unit (ReLU) nets for very high dimensional dataset under very mild assumptions. A major ingredient of our analysis is a fine-grained analysis of random activation matrices. The essential virtue of dissecting activation matrices is that it bridges the dynamics of optimization and angular distribution in high-dimensional data space. This angle-based detailed analysis leads to asymptotic characterizations of gr…
Recent theoretical work has demonstrated that deep neural networks have superior performance over shallow networks, but their training is more difficult, e.g., they suffer from the vanishing gradient problem. This problem can be typically resolved by the rectified linear unit (ReLU) activation. However, here we show that even for such activation, deep and narrow neural networks (NNs) will converge to erroneous mean or median states of the target function depending on the loss with high probability. Deep and narrow NNs are encountered in solving partial differential equations with high-order de…
Shanghai Jiao Tong University（上海交通大学）；Yitu Tech（依图科技）；University College London（伦敦大学学院）
Class labels have been empirically shown useful in improving the sample quality of generative adversarial nets (GANs). In this paper, we mathematically study the properties of the current variants of GANs that make use of class label information. With class aware gradient and cross-entropy decomposition, we reveal how class labels and associated losses influence GAN's training. Based on that, we propose Activation Maximization Generative Adversarial Networks (AM-GAN) as an advanced solution. Comprehensive experiments have been conducted to validate our analysis and evaluate the effectiveness o…
Fraunhofer IAIS（弗劳恩霍夫应用信息技术研究所）；IIT, Bombay（孟买印度理工学院）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
A recent line of work has established intriguing connections between the generalization/compression properties of a deep neural network (DNN) model and the so-called layer weights' stable ranks. Intuitively, the latter are indicators of the effective number of parameters in the net. In this work, we address some natural questions regarding the space of DNNs conditioned on the layers' stable rank, where we study feed-forward dynamics, initialization, training and expressivity. To this end, we first propose a random DNN model with a new sampling scheme based on stable rank. Then, we show how fee…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；University of California, Irvine（加利福尼亚大学尔湾分校）；Qualcomm AI Research（高通人工智能研究院）
Training activation quantized neural networks involves minimizing a piecewise constant function whose gradient vanishes almost everywhere, which is undesirable for the standard back-propagation or chain rule. An empirical way around this issue is to use a straight-through estimator (STE) (Bengio et al., 2013) in the backward pass only, so that the "gradient" through the modified chain rule becomes non-trivial. Since this unusual "gradient" is certainly not the gradient of loss function, the following question arises: why searching in its negative direction minimizes the training loss? In this …
Bosch Center for Artificial Intelligence（博世人工智能中心）
Neural Ordinary Differential Equations (N-ODEs) are a powerful building block for learning systems, which extend residual networks to a continuous-time dynamical system. We propose a Bayesian version of N-ODEs that enables well-calibrated quantification of prediction uncertainty, while maintaining the expressive power of their deterministic counterpart. We assign Bayesian Neural Nets (BNNs) to both the drift and the diffusion terms of a Stochastic Differential Equation (SDE) that models the flow of the activation map in time. We infer the posterior on the BNN weights using a straightforward ad…
There has been a great deal of recent interest in learning and approximation of functions that can be expressed as expectations of a given nonlinearity with respect to its random internal parameters. Examples of such representations include "infinitely wide" neural nets, where the underlying nonlinearity is given by the activation function of an individual neuron. In this paper, we bring this perspective to function representation by neural stochastic differential equations (SDEs). A neural SDE is an Itô diffusion process whose drift and diffusion matrix are elements of some parametric familie…
We identify and study two common failure modes for early training in deep ReLU nets. For each we give a rigorous proof of when it occurs and how to avoid it, for fully connected and residual architectures. The first failure mode, exploding/vanishing mean activation length, can be avoided by initializing weights from a symmetric distribution with variance 2/fan-in and, for ResNets, by correctly weighting the residual modules. We prove that the second failure mode, exponentially large variance of activation length, never occurs in residual nets once the first failure mode is avoided. In contrast…
We investigate what can be learned from translating numerical algorithms into neural networks. On the numerical side, we consider explicit, accelerated explicit, and implicit schemes for a general higher order nonlinear diffusion equation in 1D, as well as linear multigrid methods. On the neural network side, we identify corresponding concepts in terms of residual networks (ResNets), recurrent networks, and U-nets. These connections guarantee Euclidean stability of specific ResNets with a transposed convolution layer structure in each block. We present three numerical justifications for skip c…
Princeton University（普林斯顿大学）；Institute for Advanced Study（高等研究院）；Carnegie Mellon University（卡内基梅隆大学）
Recent works have cast some light on the mystery of why deep nets fit any data and generalize despite being very overparametrized. This paper analyzes training and generalization for a simple 2-layer ReLU net with random initialization, and provides the following improvements over recent works: (i) Using a tighter characterization of training speed than recent papers, an explanation for why training a neural net with random labels leads to slower training, as originally observed in [Zhang et al. ICLR'17]. (ii) Generalization bound independent of network size, using a data-dependent complexity …
University of California, Santa Barbara（加州大学圣巴巴拉分校）；University of California, Riverside（加州大学河滨分校）；California Polytechnic State University（加州州立理工大学）；Michigan State University（密歇根州立大学）
As a counterpoint to classical stochastic particle methods for linear diffusion equations, we develop a deterministic particle method for the weighted porous medium equation (WPME) and prove its convergence on bounded time intervals. This generalizes related work on blob methods for unweighted porous medium equations. From a numerical analysis perspective, our method has several advantages: it is meshfree, preserves the gradient flow structure of the underlying PDE, converges in arbitrary dimension, and captures the correct asymptotic behavior in simulations. That our method succeeds in captur…
ChebNet: Efficient and Stable Constructions of Deep Neural Networks with Rectified Power Units via Chebyshev Approximations
Software Development Center, Industrial and Commercial Bank of China（中国工商银行软件开发中心）；Hisilicon Semiconductor and Component Business Dept.(2012 Labs), Huawei Technologies Co., Ltd（华为技术有限公司半导体及组件业务部）；NCMIS & LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Beijing（数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
